Proceedings of the Eighth Annual ACM Symposium on
Parallel Algorithms and Architectures , June 1996, pp. 131-141.
To appear, Journal of Computer and Systems Sciences.

d>This paper analyzes the effect of virtual channels on the
performance of wormhole routing algorithms. We show that in a network
in which each edge can emulate up to q virtual channels, it is
possible to route any set of b-bit messages whose paths have
congestion c and dilation d in (b+d) c (d log d)^(1/q) 2^O(log^*
(c/d)) bit-steps. We also prove a nearly matching lower bound, i.e.,
for any values of c, d, q, and b, where c = d and b > d, we show how
to construct a network and a set of b-bit messages whose paths have
congestion c and dilation d that require Omega(bcd^(1/q)) bit-steps to
route. These bounds imply that increasing the queuing capacity q of
each edge can speed up a wormhole routing algorithm by a superlinear
factor. We also present a simple randomized wormhole routing
algorithm for the butterfly network. The algorithm routes a
k-relation on the inputs and outputs of an N-input butterfly in
O(bq(k+log N)(log^(1/q) N) log log Nk) bit-steps. We present a nearly
matching lower bound that holds for a broad class of algorithms.